Advanced Quantum Mechanics Schwabl · Problem 6.15

The Coulomb Potential

Physics — Quantum Mechanics · spin-projection-operator-dirac

Problem

(i) Given the definition

P(n) = \frac{1}{2}(1 + \gamma_5 \not{n})

, show that, under the assumptions

n^2 = -1

and

n_\mu k^\mu = 0

, the following relations are satisfied ((a))

[\Lambda_\pm(k), P(n)] = 0

\,, ((b))

\Lambda_+(k)P(n) + \Lambda_-(k)P(n) + \Lambda_+(k)P(-n) + \Lambda_-(k)P(-n) = 1

\,, ((c))

\mathrm{Tr}\,[\Lambda_\pm(k)P(\pm n)] = 1

\,, ((d))

P(n)^2 = P(n)

(ii) Consider the special case

n = (0, \hat{e}_z)

where

P(n) = \frac{1}{2}\begin{pmatrix} 1+\sigma^3 & 0 \\ 0 & 1-\sigma^3 \end{pmatrix}